What is the opposite of "eschatology"?

Eschatology has to do more with the "end of humankind" specifically than the "end of things". So, if Eschatology is concerned with the final events of history, or the ultimate destiny of humanity, then Anthropogeny is concerned with study of human origins.

Anthropogeny comes from Greek anthropo- (human) + Greek -geny (birth, origin).

But, if you want to take it in more general sense i.e. "final events of history", as history¹ can be of humans, earth, or cosmos, then it is Cosmogony.

The word comes from the Koine Greek κοσμογονία (from κόσμος "cosmos, the world") and the root of γί(γ)νομαι / γέγονα ("come into a new state of being"). In astronomy, cosmogony refers to the study of the origin of particular astrophysical objects or systems, and is most commonly used in reference to the origin of the Universe, the Solar System, or the Earth–Moon system.

And on the theological side, you can consider Theogony: "the genealogy or birth of the gods."

_{1. 2nd definition of history according to Oxford Dictionary: The whole series of past events connected with a particular person or thing.}

Where eschatos means last, protos means first, so the word is protology.

Protology

noun

The study or science of origins.

The word "eschatology" is most often used in the theological sense. Similarly, the word "genesis" is frequently used in the theological sense.

However, "eschatology" (in a general sense) of the word is the "study of last (or end) things". The corresponding opposite general term for the beginning (or origin) of anything would be "genesis". Therefore, the opposite of eschatology (in a general sense) would be "genesis studies" or "the study of geneses". This general definition being "the study of beginnings (geneses or origins)".

There are already some good answers, but none mentions the first word which comes to my mind: aetiology (alternative spelling etiology is preferred by some people).

This is the study of the origins of things in terms of causation (etymologically, the study of causes). The noun can also be used in the sense of "origin story"; this meaning is common in medicine, and fairly common in theology.

See: Wikipedia:Etiology.

American Heritage Dictionary:

n. pl. e·ti·ol·o·gies also ae·ti·ol·o·gies

1. a. The study of causes or origins.
   b. The branch of medicine that deals with the causes or origins of disease.
2. a. Assignment of a cause, an origin, or a reason for something.
   b. The cause or origin of a disease or disorder as determined by medical diagnosis.

Just to be clear on the meaning of ‘eschatology’, here it is in the ‘Oxford Dictionary of Philosophy’:

(Greek, eschatos, the last) The formation of ideas about the end of life, or the end of the world, and in Christian theology, the last judgement and resurrection.